[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121079-en":3,"doc-seo-121079-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},121079,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Analytical results for uncertainty propagation through trained machine learning regression models","Machine learning regression models are increasingly used in metrology, but credibility requires principled uncertainty quantification. This paper derives analytical expressions for the mean and variance of model outputs under specified input distributions and for multiple trained/fixed regression model classes. It covers linear regression, penalised linear regression, kernel ridge regression, Gaussian Processes, support vector machines, and relevance vector machines, and validates results with numerical experiments and a Monte Carlo comparison. A metrology example models lithium-ion state-of-health from EIS data.","arXiv :2404 . 11224v2 [ cs .LG] 8 May 2024  \nAnalytical results for uncertainty propagation through trained machine learning regression models  \nAndrew Thompson∗  \nNational Physical Laboratory, Hampton Road, Teddington, TW11 0LW, UK  \nMay 9, 2024  \nAbstract  \nMachine learning (ML) models are increasingly being used in metrology applications. However, for ML models to be credible in a metrology context they should be accompanied by principled uncertainty quantification. This paper addresses the challenge of uncertainty propagation through trained/fixed ML regression models. Analytical expressions for the mean and variance of the model output are obtained/presented for certain input data distributions and for a variety of ML models. Our results cover several popular ML models including linear regression, penalised linear regression, kernel ridge regression, Gaussian Processes (GPs), support vector machines (SVMs) and relevance vector machines (RVMs) . We present numerical experiments in which we validate our methods and compare them with a Monte Carlo approach from a computational efficiency point of view. We also illustrate our methods in the context of a metrology application, namely modelling the state-of-health of lithium-ion cells based upon Electrical Impedance Spectroscopy (EIS) data.  \nKeywords—Machine learning, uncertainty, regression, kernel models, propagation, lithium-ion cells  \n1 Introduction  \nMachine learning (ML) measurement models are increasingly being used in a wide range of metrology applications, for example in thermometry [1], battery state-of-health modelling [2], nanoparticle characterisation [3], earth observation [4] and oceanography [5] . The appeal of ML is its ability to learn measurement models from data, even in cases where physical models are either not well understood or inefficient to compute. It also enables the automation of time-consuming processes that previously needed to be performed manually.  \n∗ email: [andrew.thompson@npl.co.uk](andrew.thompson@npl.co.uk)  \n© 2024. This manuscript version is made available under the CC-BY-NC-ND 4.0 license [https://creativecommons.org/licenses/by-nc-nd/4.0/](https://creativecommons.org/licenses/by-nc-nd/4.0/)  \n1.1 Uncertainty quantification for ML models  \nFor ML approaches to be credible in a metrology context, it is important that they are accompanied by principled uncertainty quantification. The importance of uncertainty quantification for ML and the need for research on this topic was highlighted in the recent Strategic Research Agenda developed by the European Metrology Network for Mathematics and Statistics (MATHMET) [6, Section 3] . This paper addresses the challenge of uncertainty propagation through trained/fixed ML regression models. In other words, given an observation of the input variables accompanied by knowledge of their corresponding uncertainty, the aim is to characterise the uncertainty of the ML model output. We choose to focus on regression models because they occur particularly frequently in a metrology context.  \nA framework for evaluating uncertainties by means of measurement models was standardised for the metrology community in the influential “Guide to the Expression of Uncertainty in Measurement” (GUM) and related supplements [7, 8, 9, 10] . The GUM paradigm mainly addresses the propagation of uncertainty through fixed models. However, since an ML model is learned from data, the absence of full physical insight leads to uncertainty in the model itself. Indeed, two types of uncertainty are typically distinguished in the ML community: data uncertainty and model uncertainty. Data uncertainty is inherent to the modelling task, and so is in this sense irreducible. Model uncertainty, on the other hand, is uncertainty concerning the model itself, for example the form of the model or the value of its parameters. The need to take account of model uncertainty is in fact explicitly acknowledged within the GUM documents; see [8, 3.1.6] ","cbCainjOX5Ia7KXb","https://ap.wps.com/l/cbCainjOX5Ia7KXb","pdf",1325133,1,28,"English","en",105,"# Introduction\n## Uncertainty quantification for ML models\n## Analytical approaches for uncertainty quantification","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It addresses how to propagate input uncertainty through trained or fixed machine learning regression models, yielding analytical characterisations of the output uncertainty.\"},{\"question\":\"Which regression model types are covered by the analytical results?\",\"answer\":\"The paper includes linear regression, penalised linear regression, kernel ridge regression, Gaussian Processes, support vector machines, and relevance vector machines.\"},{\"question\":\"How are the proposed methods validated and compared?\",\"answer\":\"They are validated using numerical experiments and compared with a Monte Carlo approach from the perspective of computational efficiency.\"}]","Analytical results for uncertainty propagation through trained machine learning regression models | 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problem does the paper address?","Question",{"text":75,"@type":76},"It addresses how to propagate input uncertainty through trained or fixed machine learning regression models, yielding analytical characterisations of the output uncertainty.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which regression model types are covered by the analytical results?",{"text":80,"@type":76},"The paper includes linear regression, penalised linear regression, kernel ridge regression, Gaussian Processes, support vector machines, and relevance vector machines.",{"name":82,"@type":73,"acceptedAnswer":83},"How are the proposed methods validated and compared?",{"text":84,"@type":76},"They are validated using numerical experiments and compared with a Monte Carlo approach from the perspective of computational 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